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Chapter 1: Problem 17

Determine if the following pairs of angles are coterminal. \(\pi / 2\) and \(3 \pi / 2\)

Short Answer

Expert verified
The angles \(\pi / 2\) and \(3 \pi / 2\) are not coterminal since their difference (\(\pi\)) is not a multiple of \(2\pi\).

Step by step solution

01

- Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides when drawn in standard position. In other words, they differ by an integer multiple of a full rotation (which is 360 degrees or \(2\pi\) radians).
02

- Find Difference Between the Two Angles

Calculate the difference between the two given angles: \(3\pi / 2 - \pi / 2\).
03

- Simplify the Difference

Simplify the difference: \[ \frac{3\pi}{2} - \frac{\pi}{2} = \frac{3\pi - \pi}{2} = \frac{2\pi}{2} = \pi \].
04

- Compare the Result to Multiples of \(2\pi\)

Since the difference \(\pi\) is not an integer multiple of \(2\pi\), the angles \(\pi / 2\) and \(3 \pi / 2\) are not coterminal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle Difference
The concept of angle difference is essential in determining whether two angles are coterminal. To check if two angles are coterminal, we need to calculate the difference between them. For instance, in the case of \(\frac{\pi}{2}\) and \(3\pi/2\), the difference is found by subtracting one angle from the other. This difference helps us understand if the two angles differ by a full rotation (or multiple rotations), which is necessary for coterminal angles.

A common mistake is to assume angles are coterminal if their difference is significant without checking if it matches up with a full rotation, 360 degrees, or other multiples.
Radians
Radians are another way to measure angles, often used in higher mathematics and science. One full rotation (360 degrees) equals \(2\pi\) radians.

Understanding radians will help you figure out coterminal angles more efficiently.
In the example exercise, \( \frac{\pi}{2} \) and \( 3\pi/2 \), we use radians to measure the angles and find their difference. After detecting the difference, we compare it to multiples of \(2\pi\) to see if the angles are coterminal.
Using radians can simplify calculations and provide better accuracy, especially when dealing with trigonometric functions.
Full Rotation
A full rotation in terms of angle measurement is 360 degrees or \(2\pi\) radians.

When determining if two angles are coterminal, it's crucial to compare their difference to a full rotation.
If the difference between two angles is an integer multiple of \(2\pi\), they are coterminal.

From the exercise:
  • Calculate the difference: \ \frac{3\pi}{2} - \frac{\pi}{2} = \pi \
  • \(\pi\) is not an integer multiple of \(2\pi\), so the angles are not coterminal.

Mastering this concept helps with trigonometry and other math problems involving rotational symmetry.

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Most popular questions from this chapter

Graph the following functions on \([-2 \pi, 2 \pi]\). a) \(y=2 \cos 2 t-1\) b) \(y=2 \cos (2[t+\pi / 6])-1\) c) How does the horizontal shift change the graph?

Use a calculator to evaluate the following trigonometric functions. $$ \sin 164^{\circ} $$

Use a calculator to find the degree measure of an acute angle whose trigonometric function is given. \(\tan t=0.84\)

Use a calculator to evaluate the following trigonometric functions. $$ \tan \left(-220^{\circ}\right) $$

Use a calculator to find the radian measure of an acute angle whose trigonometric function is given. \(\cos t=0.60\)
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